That diagram is showing something related, but not really directly applicable. The important extra feature the controller has is the ability to add energy to t

That diagram is showing something related, but not really directly applicable. The important extra feature the controller has is the ability to add energy to the pendulum - which is akin to being able to move between locations on the plot. However, the graph doesn't suggest anything about the results of any particular move by the controller, so (even using an extended version of the plot, dealing with upright target positions) this doesn't really a way to solve for the controller's actions. What the plot shows is how complex the behaviour of a double pendulum is : small changes can result in wildly different outcomes, even though it's deterministic.